Step |
Hyp |
Ref |
Expression |
1 |
|
mreexd.1 |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
2 |
|
mreexd.2 |
⊢ ( 𝜑 → ∀ 𝑠 ∈ 𝒫 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ ( ( 𝑁 ‘ ( 𝑠 ∪ { 𝑦 } ) ) ∖ ( 𝑁 ‘ 𝑠 ) ) 𝑦 ∈ ( 𝑁 ‘ ( 𝑠 ∪ { 𝑧 } ) ) ) |
3 |
|
mreexd.3 |
⊢ ( 𝜑 → 𝑆 ⊆ 𝑋 ) |
4 |
|
mreexd.4 |
⊢ ( 𝜑 → 𝑌 ∈ 𝑋 ) |
5 |
|
mreexd.5 |
⊢ ( 𝜑 → 𝑍 ∈ ( 𝑁 ‘ ( 𝑆 ∪ { 𝑌 } ) ) ) |
6 |
|
mreexd.6 |
⊢ ( 𝜑 → ¬ 𝑍 ∈ ( 𝑁 ‘ 𝑆 ) ) |
7 |
1 3
|
sselpwd |
⊢ ( 𝜑 → 𝑆 ∈ 𝒫 𝑋 ) |
8 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑠 = 𝑆 ) → 𝑌 ∈ 𝑋 ) |
9 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑠 = 𝑆 ) ∧ 𝑦 = 𝑌 ) → 𝑍 ∈ ( 𝑁 ‘ ( 𝑆 ∪ { 𝑌 } ) ) ) |
10 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑠 = 𝑆 ) ∧ 𝑦 = 𝑌 ) → 𝑠 = 𝑆 ) |
11 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑠 = 𝑆 ) ∧ 𝑦 = 𝑌 ) → 𝑦 = 𝑌 ) |
12 |
11
|
sneqd |
⊢ ( ( ( 𝜑 ∧ 𝑠 = 𝑆 ) ∧ 𝑦 = 𝑌 ) → { 𝑦 } = { 𝑌 } ) |
13 |
10 12
|
uneq12d |
⊢ ( ( ( 𝜑 ∧ 𝑠 = 𝑆 ) ∧ 𝑦 = 𝑌 ) → ( 𝑠 ∪ { 𝑦 } ) = ( 𝑆 ∪ { 𝑌 } ) ) |
14 |
13
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑠 = 𝑆 ) ∧ 𝑦 = 𝑌 ) → ( 𝑁 ‘ ( 𝑠 ∪ { 𝑦 } ) ) = ( 𝑁 ‘ ( 𝑆 ∪ { 𝑌 } ) ) ) |
15 |
9 14
|
eleqtrrd |
⊢ ( ( ( 𝜑 ∧ 𝑠 = 𝑆 ) ∧ 𝑦 = 𝑌 ) → 𝑍 ∈ ( 𝑁 ‘ ( 𝑠 ∪ { 𝑦 } ) ) ) |
16 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑠 = 𝑆 ) ∧ 𝑦 = 𝑌 ) → ¬ 𝑍 ∈ ( 𝑁 ‘ 𝑆 ) ) |
17 |
10
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑠 = 𝑆 ) ∧ 𝑦 = 𝑌 ) → ( 𝑁 ‘ 𝑠 ) = ( 𝑁 ‘ 𝑆 ) ) |
18 |
16 17
|
neleqtrrd |
⊢ ( ( ( 𝜑 ∧ 𝑠 = 𝑆 ) ∧ 𝑦 = 𝑌 ) → ¬ 𝑍 ∈ ( 𝑁 ‘ 𝑠 ) ) |
19 |
15 18
|
eldifd |
⊢ ( ( ( 𝜑 ∧ 𝑠 = 𝑆 ) ∧ 𝑦 = 𝑌 ) → 𝑍 ∈ ( ( 𝑁 ‘ ( 𝑠 ∪ { 𝑦 } ) ) ∖ ( 𝑁 ‘ 𝑠 ) ) ) |
20 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 = 𝑆 ) ∧ 𝑦 = 𝑌 ) ∧ 𝑧 = 𝑍 ) → 𝑦 = 𝑌 ) |
21 |
|
simpllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 = 𝑆 ) ∧ 𝑦 = 𝑌 ) ∧ 𝑧 = 𝑍 ) → 𝑠 = 𝑆 ) |
22 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 = 𝑆 ) ∧ 𝑦 = 𝑌 ) ∧ 𝑧 = 𝑍 ) → 𝑧 = 𝑍 ) |
23 |
22
|
sneqd |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 = 𝑆 ) ∧ 𝑦 = 𝑌 ) ∧ 𝑧 = 𝑍 ) → { 𝑧 } = { 𝑍 } ) |
24 |
21 23
|
uneq12d |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 = 𝑆 ) ∧ 𝑦 = 𝑌 ) ∧ 𝑧 = 𝑍 ) → ( 𝑠 ∪ { 𝑧 } ) = ( 𝑆 ∪ { 𝑍 } ) ) |
25 |
24
|
fveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 = 𝑆 ) ∧ 𝑦 = 𝑌 ) ∧ 𝑧 = 𝑍 ) → ( 𝑁 ‘ ( 𝑠 ∪ { 𝑧 } ) ) = ( 𝑁 ‘ ( 𝑆 ∪ { 𝑍 } ) ) ) |
26 |
20 25
|
eleq12d |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 = 𝑆 ) ∧ 𝑦 = 𝑌 ) ∧ 𝑧 = 𝑍 ) → ( 𝑦 ∈ ( 𝑁 ‘ ( 𝑠 ∪ { 𝑧 } ) ) ↔ 𝑌 ∈ ( 𝑁 ‘ ( 𝑆 ∪ { 𝑍 } ) ) ) ) |
27 |
19 26
|
rspcdv |
⊢ ( ( ( 𝜑 ∧ 𝑠 = 𝑆 ) ∧ 𝑦 = 𝑌 ) → ( ∀ 𝑧 ∈ ( ( 𝑁 ‘ ( 𝑠 ∪ { 𝑦 } ) ) ∖ ( 𝑁 ‘ 𝑠 ) ) 𝑦 ∈ ( 𝑁 ‘ ( 𝑠 ∪ { 𝑧 } ) ) → 𝑌 ∈ ( 𝑁 ‘ ( 𝑆 ∪ { 𝑍 } ) ) ) ) |
28 |
8 27
|
rspcimdv |
⊢ ( ( 𝜑 ∧ 𝑠 = 𝑆 ) → ( ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ ( ( 𝑁 ‘ ( 𝑠 ∪ { 𝑦 } ) ) ∖ ( 𝑁 ‘ 𝑠 ) ) 𝑦 ∈ ( 𝑁 ‘ ( 𝑠 ∪ { 𝑧 } ) ) → 𝑌 ∈ ( 𝑁 ‘ ( 𝑆 ∪ { 𝑍 } ) ) ) ) |
29 |
7 28
|
rspcimdv |
⊢ ( 𝜑 → ( ∀ 𝑠 ∈ 𝒫 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ ( ( 𝑁 ‘ ( 𝑠 ∪ { 𝑦 } ) ) ∖ ( 𝑁 ‘ 𝑠 ) ) 𝑦 ∈ ( 𝑁 ‘ ( 𝑠 ∪ { 𝑧 } ) ) → 𝑌 ∈ ( 𝑁 ‘ ( 𝑆 ∪ { 𝑍 } ) ) ) ) |
30 |
2 29
|
mpd |
⊢ ( 𝜑 → 𝑌 ∈ ( 𝑁 ‘ ( 𝑆 ∪ { 𝑍 } ) ) ) |